Mathematics 11–12 · Year 12

Filling vessels at a steady rate: height-time graphs for a cylinder and a cone

Algebraic relationships: Graphs of practical situations (Mathematics Standard 1, Year 12: the cylinder's straight line and the description of the situation only, because that content group covers linear and exponential graphs and the cone's height-time graph is neither); Applications of calculus: Rates of change (Mathematics Advanced, Year 12: the rate of rise as the derivative of height with respect to time; the chain rule itself is Mathematics Advanced Year 11, and the related-rates step for the cone is Extension 1); Further applications of calculus: Further rates of change (Mathematics Extension 1, Year 12: related rates through the chain rule)

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

The same steady inflow gives a straight height-time graph in a cylinder and a curved one in a cone, because the rate of rise depends on the surface area at the current height.

What you need

A straight-sided 1 L jar or measuring cylinder and a funnel or cone-shaped vessel with its outlet sealed
A tap set to a slow, steady flow that takes at least a minute to fill the 1 L vessel (below about 17 mL/s); measure the flow with a jug and stopwatch first
A ruler taped to each vessel, a stopwatch

How to do it

Measure the flow rate Q by timing 500 mL into a jug.
Fill the cylinder from the steady source and record the height every 10 s until full.
Repeat for the cone-shaped vessel.
Plot both height-time graphs on one set of axes and describe each shape.
For the cylinder compute the predicted rate of rise Q / (pi r^2) and compare it with the gradient; for the cone (Extension 1) use V = pi tan^2(alpha) h^3 / 3 and the chain rule to predict dh/dt at two heights.

What you should see

Model check at the simulation's default Q = 100 mL/s: in a cylinder of radius 5.0 cm the water rises at a steady 1.27 cm/s, a straight line. Into a cone with its apex down and radius equal to half its height, the height is 12.4 cm after 500 mL and 15.6 cm after 1000 mL, rising at 0.83 cm/s and 0.52 cm/s: a curve that starts steep and flattens. At the learner's slower tap every rate scales by Q / 100 and the shapes are the same. The learner knows it worked when the cylinder's points lie on a line whose gradient matches Q / (pi r^2) within 10 per cent.

What changes

What you change: time
What you measure: water height
What you keep the same: flow rate Q
vessel
reading at eye level

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

Water rises at the same speed in any container fed by the same tap; the speed depends on the cross-section at the surface.
A curved graph means the tap was turned; the inflow was constant and the shape of the vessel made the curve.

Safety card

Low riskLearners carry it out

Hazards

water on the floor

Controls

work in a sink or over a tray

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 1 focus area Algebraic relationships; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S1-01
Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Applications of calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-06
Mathematics Extension 1 11–12 Syllabus (2024), Year 12 focus area Further applications of calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-12-05
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities